Species and Hyperplane Arrangements
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This dissertation has two leading characters: Hopf monoids in the category of species and the Tits algebra of a real hyperplane arrangement. The relation between these two comes from the work of Aguiar and Mahajan (2013), who showed that a (co)commutative Hopf monoid gives rise to a family of (left)right-modules over the Tits algebra of the braid arrangement in all dimensions. One goal of this thesis is to explore the representation theory of the Tits algebra of arbitrary affine arrangements to extend what is known in the case of linear arrangements and to give an insight into some unanswered questions in the field of Hopf monoids. In the first part, we extend the study of characteristic elements of a hyperplane arrangement from the linear to the affine case. We present the basic properties of these elements and apply them to derive numerous results about the characteristic polynomial of an arrangement, from Zaslavsky's formulas to more recent results of Kung and of Klivans and Swartz. We construct several examples of characteristic elements, including one in terms of intrinsic volumes of faces of the arrangement. In the second part, we study deformations
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Stillman, Michael